Graph minors. II. Algorithmic aspects of tree-width

1986 ◽  
Vol 7 (3) ◽  
pp. 309-322 ◽  
Author(s):  
Neil Robertson ◽  
P.D Seymour
Keyword(s):  
1984 ◽  
Vol 36 (1) ◽  
pp. 49-64 ◽  
Author(s):  
Neil Robertson ◽  
P.D Seymour
Keyword(s):  

2021 ◽  
Author(s):  
◽  
Jeffrey Donald Azzato

<p>It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids.  Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width.</p>


1990 ◽  
Vol 48 (2) ◽  
pp. 227-254 ◽  
Author(s):  
Neil Robertson ◽  
P.D Seymour

2021 ◽  
Author(s):  
◽  
Jeffrey Donald Azzato

<p>It is natural to try to extend the results of Robertson and Seymour's Graph Minors Project to other objects. As linked tree-decompositions (LTDs) of graphs played a key role in the Graph Minors Project, establishing the existence of ltds of other objects is a useful step towards such extensions. There has been progress in this direction for both infinite graphs and matroids.  Kris and Thomas proved that infinite graphs of finite tree-width have LTDs. More recently, Geelen, Gerards and Whittle proved that matroids have linked branch-decompositions, which are similar to LTDs. These results suggest that infinite matroids of finite treewidth should have LTDs. We answer this conjecture affirmatively for the representable case. Specifically, an independence space is an infinite matroid, and a point configuration (hereafter configuration) is a represented independence space. It is shown that every configuration having tree-width has an LTD k E w (kappa element of omega) of width at most 2k. Configuration analogues for bridges of X (also called connected components modulo X) and chordality in graphs are introduced to prove this result. A correspondence is established between chordal configurations only containing subspaces of dimension at most k E w (kappa element of omega) and configuration tree-decompositions having width at most k. This correspondence is used to characterise finite-width LTDs of configurations by their local structure, enabling the proof of the existence result. The theory developed is also used to show compactness of configuration tree-width: a configuration has tree-width at most k E w (kappa element of omega) if and only if each of its finite subconfigurations has tree-width at most k E w (kappa element of omega). The existence of LTDs for configurations having finite tree-width opens the possibility of well-quasi-ordering (or even better-quasi-ordering) by minors those independence spaces representable over a fixed finite field and having bounded tree-width.</p>


Agronomy ◽  
2019 ◽  
Vol 9 (11) ◽  
pp. 741 ◽  
Author(s):  
Haihui Yang ◽  
Xiaochan Wang ◽  
Guoxiang Sun

Perception of the fruit tree canopy is a vital technology for the intelligent control of a modern standardized orchard. Due to the complex three-dimensional (3D) structure of the fruit tree canopy, morphological parameters extracted from two-dimensional (2D) or single-perspective 3D images are not comprehensive enough. Three-dimensional information from different perspectives must be combined in order to perceive the canopy information efficiently and accurately in complex orchard field environment. The algorithms used for the registration and fusion of data from different perspectives and the subsequent extraction of fruit tree canopy related parameters are the keys to the problem. This study proposed a 3D morphological measurement method for a fruit tree canopy based on Kinect sensor self-calibration, including 3D point cloud generation, point cloud registration and canopy information extraction of apple tree canopy. Using 32 apple trees (Yanfu 3 variety) morphological parameters of the height (H), maximum canopy width (W) and canopy thickness (D) were calculated. The accuracy and applicability of this method for extraction of morphological parameters were statistically analyzed. The results showed that, on both sides of the fruit trees, the average relative error (ARE) values of the morphological parameters including the fruit tree height (H), maximum tree width (W) and canopy thickness (D) between the calculated values and measured values were 3.8%, 12.7% and 5.0%, respectively, under the V1 mode; the ARE values under the V2 mode were 3.3%, 9.5% and 4.9%, respectively; and the ARE values under the V1 and V2 merged mode were 2.5%, 3.6% and 3.2%, respectively. The measurement accuracy of the tree width (W) under the double visual angle mode had a significant advantage over that under the single visual angle mode. The 3D point cloud reconstruction method based on Kinect self-calibration proposed in this study has high precision and stable performance, and the auxiliary calibration objects are readily portable and easy to install. It can be applied to different experimental scenes to extract 3D information of fruit tree canopies and has important implications to achieve the intelligent control of standardized orchards.


2002 ◽  
Vol 11 (6) ◽  
pp. 541-547 ◽  
Author(s):  
PATRICK BELLENBAUM ◽  
REINHARD DIESTEL

We give short proofs of the following two results: Thomas's theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width; and the ‘tree-width duality theorem’ of Seymour and Thomas, that the tree-width of a finite graph is exactly one less than the largest order of its brambles.


2007 ◽  
Vol 51 (3) ◽  
pp. 326-362 ◽  
Author(s):  
P. Hlineny ◽  
S.-i. Oum ◽  
D. Seese ◽  
G. Gottlob
Keyword(s):  

Sign in / Sign up

Export Citation Format

Share Document